Comprehensive Class 8 math worksheet designed to help students master solving linear equations in one variable.

Let's solve each of the given linear equations step by step. The goal is to isolate the variable on one side of the equation.

---

1. Solve:

\[
\frac{2x + 5}{3} = 3x - 10
\]

#### Step 1: Eliminate the fraction by multiplying both sides by 3:
\[
3 \cdot \frac{2x + 5}{3} = 3 \cdot (3x - 10)
\]
\[
2x + 5 = 9x - 30
\]

#### Step 2: Move all terms involving \(x\) to one side and constants to the other:
\[
2x - 9x = -30 - 5
\]
\[
-7x = -35
\]

#### Step 3: Solve for \(x\):
\[
x = \frac{-35}{-7}
\]
\[
x = 5
\]

Answer:
\[
\boxed{5}
\]

---

2. Solve:

\[
\frac{a - 8}{3} = \frac{a - 3}{2}
\]

#### Step 1: Eliminate the fractions by finding the least common denominator (LCD), which is 6. Multiply both sides by 6:
\[
6 \cdot \frac{a - 8}{3} = 6 \cdot \frac{a - 3}{2}
\]
\[
2(a - 8) = 3(a - 3)
\]

#### Step 2: Distribute the constants:
\[
2a - 16 = 3a - 9
\]

#### Step 3: Move all terms involving \(a\) to one side and constants to the other:
\[
2a - 3a = -9 + 16
\]
\[
-a = 7
\]

#### Step 4: Solve for \(a\):
\[
a = -7
\]

Answer:
\[
\boxed{-7}
\]

---

3. Solve:

\[
\frac{7y + 2}{5} = \frac{6y - 5}{11}
\]

#### Step 1: Eliminate the fractions by finding the LCD, which is 55. Multiply both sides by 55:
\[
55 \cdot \frac{7y + 2}{5} = 55 \cdot \frac{6y - 5}{11}
\]
\[
11(7y + 2) = 5(6y - 5)
\]

#### Step 2: Distribute the constants:
\[
77y + 22 = 30y - 25
\]

#### Step 3: Move all terms involving \(y\) to one side and constants to the other:
\[
77y - 30y = -25 - 22
\]
\[
47y = -47
\]

#### Step 4: Solve for \(y\):
\[
y = \frac{-47}{47}
\]
\[
y = -1
\]

Answer:
\[
\boxed{-1}
\]

---

4. Solve:

\[
x - 2x + 2 - \frac{16}{3x} + 5 = 3 - \frac{7}{2x}
\]

#### Step 1: Simplify the left-hand side:
\[
x - 2x + 2 + 5 - \frac{16}{3x} = -x + 7 - \frac{16}{3x}
\]

So the equation becomes:
\[
-x + 7 - \frac{16}{3x} = 3 - \frac{7}{2x}
\]

#### Step 2: Eliminate the fractions by finding the LCD, which is \(6x\). Multiply both sides by \(6x\):
\[
6x \left( -x + 7 - \frac{16}{3x} \right) = 6x \left( 3 - \frac{7}{2x} \right)
\]

Distribute \(6x\):
\[
6x(-x) + 6x(7) - 6x \cdot \frac{16}{3x} = 6x(3) - 6x \cdot \frac{7}{2x}
\]
\[
-6x^2 + 42x - 32 = 18x - 21
\]

#### Step 3: Move all terms to one side:
\[
-6x^2 + 42x - 32 - 18x + 21 = 0
\]
\[
-6x^2 + 24x - 11 = 0
\]

#### Step 4: Solve the quadratic equation using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = -6\), \(b = 24\), and \(c = -11\):
\[
x = \frac{-24 \pm \sqrt{24^2 - 4(-6)(-11)}}{2(-6)}
\]
\[
x = \frac{-24 \pm \sqrt{576 - 264}}{-12}
\]
\[
x = \frac{-24 \pm \sqrt{312}}{-12}
\]
\[
x = \frac{-24 \pm 2\sqrt{78}}{-12}
\]
\[
x = \frac{-12 \pm \sqrt{78}}{-6}
\]
\[
x = 2 \mp \frac{\sqrt{78}}{6}
\]

The solutions are:
\[
x = 2 - \frac{\sqrt{78}}{6} \quad \text{or} \quad x = 2 + \frac{\sqrt{78}}{6}
\]

Since the problem asks for proper fractions, we need to check if these can be simplified further. However, they are already in their simplest forms.

Answer:
\[
\boxed{2 - \frac{\sqrt{78}}{6}, 2 + \frac{\sqrt{78}}{6}}
\]

---

5. Solve:

\[
\frac{1}{2}x + 7x - 6 = 7x + \frac{1}{4}
\]

#### Step 1: Combine like terms on the left-hand side:
\[
\frac{1}{2}x + 7x - 6 = \frac{1}{2}x + 7x - 6
\]
\[
\frac{1}{2}x + 7x = \frac{1}{2}x + \frac{14}{2}x = \frac{15}{2}x
\]

So the equation becomes:
\[
\frac{15}{2}x - 6 = 7x + \frac{1}{4}
\]

#### Step 2: Eliminate the fractions by finding the LCD, which is 4. Multiply both sides by 4:
\[
4 \left( \frac{15}{2}x - 6 \right) = 4 \left( 7x + \frac{1}{4} \right)
\]
\[
4 \cdot \frac{15}{2}x - 4 \cdot 6 = 4 \cdot 7x + 4 \cdot \frac{1}{4}
\]
\[
30x - 24 = 28x + 1
\]

#### Step 3: Move all terms involving \(x\) to one side and constants to the other:
\[
30x - 28x = 1 + 24
\]
\[
2x = 25
\]

#### Step 4: Solve for \(x\):
\[
x = \frac{25}{2}
\]

Answer:
\[
\boxed{\frac{25}{2}}
\]

---

6. Solve:

\[
\frac{3}{4}x + 4x = \frac{7}{8} + 6x - 6
\]

#### Step 1: Combine like terms on both sides:
On the left-hand side:
\[
\frac{3}{4}x + 4x = \frac{3}{4}x + \frac{16}{4}x = \frac{19}{4}x
\]

On the right-hand side:
\[
\frac{7}{8} + 6x - 6 = 6x + \frac{7}{8} - \frac{48}{8} = 6x - \frac{41}{8}
\]

So the equation becomes:
\[
\frac{19}{4}x = 6x - \frac{41}{8}
\]

#### Step 2: Eliminate the fractions by finding the LCD, which is 8. Multiply both sides by 8:
\[
8 \cdot \frac{19}{4}x = 8 \cdot 6x - 8 \cdot \frac{41}{8}
\]
\[
2 \cdot 19x = 48x - 41
\]
\[
38x = 48x - 41
\]

#### Step 3: Move all terms involving \(x\) to one side and constants to the other:
\[
38x - 48x = -41
\]
\[
-10x = -41
\]

#### Step 4: Solve for \(x\):
\[
x = \frac{-41}{-10}
\]
\[
x = \frac{41}{10}
\]

Answer:
\[
\boxed{\frac{41}{10}}
\]

---

7. Solve:

\[
\frac{7x}{2} - \frac{5x}{2} = \frac{20x}{3} + 10
\]

#### Step 1: Combine like terms on the left-hand side:
\[
\frac{7x}{2} - \frac{5x}{2} = \frac{2x}{2} = x
\]

So the equation becomes:
\[
x = \frac{20x}{3} + 10
\]

#### Step 2: Eliminate the fraction by multiplying both sides by 3:
\[
3 \cdot x = 3 \cdot \left( \frac{20x}{3} + 10 \right)
\]
\[
3x = 20x + 30
\]

#### Step 3: Move all terms involving \(x\) to one side and constants to the other:
\[
3x - 20x = 30
\]
\[
-17x = 30
\]

#### Step 4: Solve for \(x\):
\[
x = \frac{30}{-17}
\]
\[
x = -\frac{30}{17}
\]

Answer:
\[
\boxed{-\frac{30}{17}}
\]

---

8. Solve:

\[
\frac{6x + 1}{2} + 1 = \frac{7x - 3}{3}
\]

#### Step 1: Eliminate the fractions by finding the LCD, which is 6. Multiply both sides by 6:
\[
6 \left( \frac{6x + 1}{2} + 1 \right) = 6 \left( \frac{7x - 3}{3} \right)
\]
\[
6 \cdot \frac{6x + 1}{2} + 6 \cdot 1 = 6 \cdot \frac{7x - 3}{3}
\]
\[
3(6x + 1) + 6 = 2(7x - 3)
\]

#### Step 2: Distribute the constants:
\[
18x + 3 + 6 = 14x - 6
\]
\[
18x + 9 = 14x - 6
\]

#### Step 3: Move all terms involving \(x\) to one side and constants to the other:
\[
18x - 14x = -6 - 9
\]
\[
4x = -15
\]

#### Step 4: Solve for \(x\):
\[
x = \frac{-15}{4}
\]

Answer:
\[
\boxed{-\frac{15}{4}}
\]

---

9. Solve:

\[
\frac{3a - 2}{3} + \frac{2a + 3}{2} = a + \frac{7}{6}
\]

#### Step 1: Eliminate the fractions by finding the LCD, which is 6. Multiply both sides by 6:
\[
6 \left( \frac{3a - 2}{3} + \frac{2a + 3}{2} \right) = 6 \left( a + \frac{7}{6} \right)
\]
\[
6 \cdot \frac{3a - 2}{3} + 6 \cdot \frac{2a + 3}{2} = 6 \cdot a + 6 \cdot \frac{7}{6}
\]
\[
2(3a - 2) + 3(2a + 3) = 6a + 7
\]

#### Step 2: Distribute the constants:
\[
6a - 4 + 6a + 9 = 6a + 7
\]
\[
12a + 5 = 6a + 7
\]

#### Step 3: Move all terms involving \(a\) to one side and constants to the other:
\[
12a - 6a = 7 - 5
\]
\[
6a = 2
\]

#### Step 4: Solve for \(a\):
\[
a = \frac{2}{6}
\]
\[
a = \frac{1}{3}
\]

Answer:
\[
\boxed{\frac{1}{3}}
\]

---

10. Solve:

\[
x - \frac{x - 1}{2} = 1 - \frac{x - 2}{3}
\]

#### Step 1: Eliminate the fractions by finding the LCD, which is 6. Multiply both sides by 6:
\[
6 \left( x - \frac{x - 1}{2} \right) = 6 \left( 1 - \frac{x - 2}{3} \right)
\]
\[
6 \cdot x - 6 \cdot \frac{x - 1}{2} = 6 \cdot 1 - 6 \cdot \frac{x - 2}{3}
\]
\[
6x - 3(x - 1) = 6 - 2(x - 2)
\]

#### Step 2: Distribute the constants:
\[
6x - 3x + 3 = 6 - 2x + 4
\]
\[
3x + 3 = 10 - 2x
\]

#### Step 3: Move all terms involving \(x\) to one side and constants to the other:
\[
3x + 2x = 10 - 3
\]
\[
5x = 7
\]

#### Step 4: Solve for \(x\):
\[
x = \frac{7}{5}
\]

Answer:
\[
\boxed{\frac{7}{5}}
\]

---

Final Answers:

\[
\boxed{5, -7, -1, 2 - \frac{\sqrt{78}}{6}, 2 + \frac{\sqrt{78}}{6}, \frac{25}{2}, \frac{41}{10}, -\frac{30}{17}, -\frac{15}{4}, \frac{1}{3}, \frac{7}{5}}
\]